error function

The error function ${\rm erf}\colon\mathbb{C}\to\mathbb{C}$ is defined as follows:

 ${\rm erf}(z)={2\over\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\,dt$

The complementary error function ${\rm erfc}\colon\mathbb{C}\to\mathbb{C}$ is defined as

 ${\rm erfc}(z)={2\over\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\,dt$

The name “error function” comes from the role that these functions play in the theory of the normal random variable. It is also worth noting that the error function is a special case of the confluent hypergeometric functions and of the Mittag-Leffler function.

Note.  By Cauchy integral theorem (http://planetmath.org/SecondFormOfCauchyIntegralTheorem), the choice path of integration in the definition of ${\rm erf}$ is irrelevant since the integrand is an entire function. In the definition of ${\rm erfc}$, the path may be taken to be a half-line parallel to the positive real axis with endpoint $z$.

Title error function ErrorFunction 2013-03-22 14:46:51 2013-03-22 14:46:51 rspuzio (6075) rspuzio (6075) 10 rspuzio (6075) Definition msc 33B20 AreaUnderGaussianCurve ListOfImproperIntegrals UsingConvolutionToFindLaplaceTransform complementary error function